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In [[homological algebra]], the '''Tor functors''' are the [[derived functor]]s of the [[tensor product]] functor. They were first defined in generality to express the [[Künneth theorem]] and [[universal coefficient theorem]] in [[algebraic topology]].{{Citation needed|date=December 2008}}
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Specifically, suppose ''R'' is a [[ring (mathematics)|ring]], and denoted by ''R''-'''Mod''' the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and by '''Mod'''-''R'' the category of right ''R''-modules (if ''R'' is [[commutative ring|commutative]], the two categories coincide). Pick a fixed module ''B'' in ''R''-'''Mod'''. For ''A'' in '''Mod'''-''R'', set ''T''(''A'') = ''A''&otimes;<sub>''R''</sub>''B''. Then ''T'' is a [[right exact functor]] from '''Mod'''-''R'' to the [[category of abelian groups]] '''Ab''' (in the case when ''R'' is commutative, it is a right exact functor from '''Mod'''-''R'' to '''Mod'''-''R'') and its [[derived functor|left derived functor]]s ''L<sub>n</sub>T'' are defined. We set
 
: <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math>
 
i.e., we take a [[Projective module#Projective resolutions|projective resolution]]
 
: <math>\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0</math>
 
then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex
 
: <math>\cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B \rightarrow P_0\otimes_R B  \rightarrow 0</math>
 
(note that ''A''&otimes;<sub>''R''</sub>''B'' does not appear and the last arrow is just the zero map) and take the [[homology (mathematics)|homology]] of this complex.  
 
== Properties ==
* For every ''n'' &ge; 1, Tor{{su|b=''n''|p=''R''}}  is an [[additive functor]] from '''Mod'''-''R'' &times; ''R''-'''Mod''' to '''Ab'''. In the case when ''R'' is commutative, we have additive functors from '''Mod'''-''R'' &times; '''Mod'''-''R'' to '''Mod'''-''R''.
 
* As is true for every family of derived functors, every [[short exact sequence]] 0 → ''K'' → ''L'' → ''M'' → 0 induces a [[long exact sequence]] of the form
::<math>\cdots\rightarrow\mathrm{Tor}_2^R(M,B)\rightarrow\mathrm{Tor}_1^R(K,B)\rightarrow\mathrm{Tor}_1^R(L,B)\rightarrow\mathrm{Tor}_1^R(M,B)\rightarrow K\otimes B\rightarrow L\otimes B\rightarrow M\otimes B\rightarrow 0</math>.
 
* If ''R'' is commutative and ''r'' in ''R'' is not a [[zero divisor]] then
::<math>\mathrm{Tor}_1^R(R/(r),B)=\{b\in B:rb=0\},</math>
from which the terminology ''Tor'' (that is, ''Torsion'') comes: see [[torsion subgroup]].
 
* Tor{{su|b=''n''|p='''Z'''}}(''A'',''B'') = 0 for all ''n'' &ge; 2. The reason: every [[abelian group]] ''A'' has a [[free resolution]] of length 1, since subgroups of [[free abelian group]]s are free abelian. So in this important special case, the higher Tor functors are invisible. In addition, Tor{{su|b=1|p='''Z'''}}('''Z'''/''k'''''Z'''</sub>,''A'') =  Ker(''f'') where ''f'' represents "multiplication by ''k''".
 
* Furthermore, every free module has a free resolution of length zero, so by the argument above, if ''F'' is a free ''R''-module, then Tor{{su|b=''n''|p=''R''}}(''F,B'') = 0 for all ''n'' &ge; 1.
 
* The Tor functors preserve [[filtered colimit]]s and arbitrary [[direct sum of modules|direct sums]]: there is a [[natural isomorphism]]
::<math>\mathrm{Tor}_n^R \left (\bigoplus_i A_i, \bigoplus_j B_j \right) \simeq \bigoplus_i \bigoplus_j \mathrm{Tor}_n^R(A_i,B_j)</math>
:
 
* From the [[Finitely generated abelian group|classification of finitely generated abelian groups]], we know that every finitely generated abelian group is the direct sum of copies of '''Z''' and '''Z'''<sub>''k''</sub>. This together with the previous three points allows us to compute Tor{{su|b=1|p='''Z'''}}(''A'', ''B'') whenever ''A'' is finitely generated.
 
* A module ''M'' in '''Mod'''-''R'' is [[flat module|flat]] if and only if Tor{{su|b=1|p=''R''}}(''M'', -) = 0. In this case, we even have Tor{{su|b=''n''|p=''R''}}(''M'', -) = 0 for all ''n'' &ge; 1 . In fact, to compute Tor{{su|b=''n''|p=''R''}}(''A,B''), one may use a ''[[flat resolution]]'' of ''A'' or ''B'', instead of a projective resolution (note that a projective resolution is automatically a flat resolution, but the converse isn't true, so allowing flat resolutions is more flexible).
 
==See also==
*[[Ext functor]]
 
==References==
* {{Weibel IHA}}
 
{{DEFAULTSORT:Tor Functor}}
[[Category:Homological algebra]]
[[Category:Binary operations]]

Revision as of 15:39, 3 March 2014

If you compare registry cleaners there are a number of factors to look out for. Because of the sheer number of for registry products accessible found on the Internet at when it could be very effortless to be scammed. Something often overlooked is that a few of these cleaners might the fact is end up damaging your PC. And the registry they state they have cleaned may just lead to more problems with the computer than the ones we started with.

You may find that there are registry products that are free plus those which we will have to pay a nominal sum for. Some registry products provide a bare bones system for free with all the way of upgrading to a more advanced, powerful adaptation of the same system.

Although this problem affects millions of computer consumers throughout the globe, there is an easy method to fix it. We see, there's one reason for a slow loading computer, plus that's considering the PC cannot read the files it must run. In a nutshell, this simply means which whenever you do anything on Windows, it requires to read up on how to do it. It's traditionally a surprisingly 'dumb' system, which has to have files to tell it to do everything.

If which refuses to function we should try plus repair the problem with a 'registry cleaner'. What occurs on many computers is the fact that their registry database becomes damaged plus unable to show a computer where the DLL files which it demands are. Every Windows PC has a central 'registry' database that shops information regarding all of the DLL files on a computer.

Another thing you should check is whether the tuneup utilities system you are considering has the ability to detect files plus programs which are superior. One of the registry cleaner programs we could try is RegCure. It is helpful for speeding up plus cleaning up difficulties on a computer.

S/w connected error handling - If the blue screen bodily memory dump arises following the installation of s/w application or perhaps a driver it may be that there is system incompatibility. By booting into safe mode and removing the software we can promptly fix this error. We may equally try out a "system restore" to revert to an earlier state.

By restoring the state of your system to an earlier date, error 1721 could not appear inside Windows 7, Vista and XP. There is a tool called System Restore that you have to use in this process.

Next, there is an easy method to deal with this problem. We can install a registry cleaner that there are it online. This software may help you find out these errors in your computer and clean them. It additionally will figure out these malware and different threats that influence the speed of the computer. So this software can speed up PC simpler. We can choose one of these methods to speed up you computer.

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